9 Chapter Notes Algebra 1

Graphing Quadratic Functions 9-1 Notes for Algebra 1 Graphing Quadratic Functions

9-1 pg. 550 22-56e, 81-92(x3)

Quadratic Functions (Standard Form 𝒂𝒙 𝟐 +𝒃𝒙+𝒄=𝟎, where 𝒂≠𝟎) The shape is Non-linear called a parabola Parabolas are symmetric about a central line called the axis of symmetry. The axis of symmetry intersects the parabola at only one point called the vertex.

Maximum and Minimum When 𝑎>0, the parabola opens up the vertex is called the minimum (because it is the lowest point). When 𝑎<0, the parabola opens down the vertex is called the maximum (because it is the highest point).

Graphing the parabolas 𝑎𝑥 2 +𝑏𝑥+𝑐=0 To find the x-coordinate of the vertex use x=− 𝑏 2𝑎 The 𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 is 𝑐. Make a table with 2 units smaller and 2 units larger than the 𝑥− 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the vertex, then solve for the 𝑦−𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠.

Example 1: Graph a parabola 1.) Use a table of values to graph 𝑦=𝑥 2 −𝑥−2. State the domain and range.

Example 1: Graph a parabola 1.) Use a table of values to graph 𝑦=𝑥 2 −𝑥−2. State the domain and range. 𝑥=− −1 2 1 D− 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 R− 𝑦 𝑦≥−2 1 4 4 3 2 1 -3 -2 -1 𝑥 𝑦 −1 −2 1 2 − 9 4 1 2

Example 2: Identify Characteristics from Graphs Find the vertex, the equation of the axis of symmetry, and y-intercept. 1.) 2.) 6 5 4 3 2 1 -4 -3 -2 -1 7

Example 2: Identify Characteristics from Graphs Find the vertex, the equation of the axis of symmetry, and y-intercept. 1.) 2.) 6 5 4 3 2 1 -4 -3 -2 -1 7 𝑥=2 2, 4 0, 2 2, −2 𝑥=2 0, −4

Example 3: Identify Characteristics from Equations Find the vertex, the equation of the axis of symmetry and y-intercept. 1.) 𝑦=−2𝑥 2 −8𝑥−2 2.) 𝑦=3𝑥 2 +6𝑥−2

Example 3: Identify Characteristics from Equations Find the vertex, the equation of the axis of symmetry and y-intercept. 1.) 𝑦=−2𝑥 2 −8𝑥−2 2.) 𝑦=3𝑥 2 +6𝑥−2 −2, 6 −1, −5 𝑥=−2 𝑥=−1 0, −2 0, −2

Example 4: Maximum and Minimum Values Consider 𝑓 𝑥 =− 𝑥 2 −2𝑥−2. a.) Determine whether the function has a maximum or a minimum value. b.) State the maximum or minimum value of the function. c.) State the domain and range of the function.

Example 4: Maximum and Minimum Values Consider 𝑓 𝑥 =− 𝑥 2 −2𝑥−2. a.) Determine whether the function has a maximum or a minimum value. 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 b.) State the maximum or minimum value of the function. −1 c.) State the domain and range of the function. 𝐷= 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 ;𝑅= 𝑦 𝑦≤−1

Example 5: Graph Quadratic Functions

Example 5: Graph Quadratic Functions 4 3 2 1 -4 -3 -2 -1

Example 6: Use a Graph of a Quadratic Function ARCHERY Ben shoots an arrow. The height of the arrow can be modeled by 𝑦=−16𝑥 2 +100𝑥+4, where 𝑦 represents the height in feet of the arrow 𝑥 seconds after it is shot into the air. 1.) Graph the height of the arrow. 2.) At what height was the arrow shot? 3.) What is the maximum height of the arrow?

Example 6: Use a Graph of a Quadratic Function ARCHERY Ben shoots an arrow. The height of the arrow can be modeled by 𝑦=−16𝑥 2 +100𝑥+4, where 𝑦 represents the height in feet of the arrow 𝑥 seconds after it is shot into the air. 1.) Graph the height of the arrow. 2.) At what height was the arrow shot? 4 𝑓𝑒𝑒𝑡 3.) What is the maximum height of the arrow? 160 1 4 𝑓𝑒𝑒𝑡 180 160 140 120 100 80 60 40 20 -1 1 2 3 4 5 6 7 8

Solving Quadratic equations by graphing. 9-2 Algebra 1 Notes Solving Quadratic equations by graphing.

9-2 pg. 558 10-26e, 57-75(x3)

Example 1: Two Roots Solve 𝑥 2 −3𝑥−10=0 by graphing. 12 10 8 6 4 2 -10 -8 -6 -4 -2 Solve 𝑥 2 −3𝑥−10=0 by graphing.

Example 1: Two Roots Solve 𝑥 2 −3𝑥−10=0 By graphing. 𝑥= − −3 2 1 𝑥= 3 2 𝑥=−2, 5 8 6 4 2 -10 -8 -6 -4 -2 10 12 -12 -14 𝒙 𝒚 −10 1 −12 3 2 −12.25 2 3

Example 2: Double Root Solve 𝑥 2 +8𝑥=−16 By graphing. 20 18 16 14 12 10 8 6 4 2 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 Solve 𝑥 2 +8𝑥=−16 By graphing.

Example 2: Double Root 20 18 16 14 12 10 8 6 4 2 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 Solve 𝑥 2 +8𝑥=−16 By graphing. 𝑥= − 8 2 1 𝑥=−4 𝑥=−4 𝒙 𝒚 −6 4 −5 1 −4 −3 −2

Example 3: No Real Roots Solve 𝑥 2 +2𝑥+3=0 by graphing. 10 9 8 7 6 5 4 -5 -4 -3 -2 -1 Solve 𝑥 2 +2𝑥+3=0 by graphing.

Example 3: No Real Roots 10 9 8 7 6 5 4 3 2 1 -5 -4 -3 -2 -1 Solve 𝑥 2 +2𝑥+3=0 by graphing. 𝑥= − 2 2 1 𝑥=−1 𝑥=∅ 𝒙 𝒚 −3 6 −2 3 −1 2 1

Example 4: Approximate Roots with a table 7 6 5 4 3 2 1 -3 -2 -1 8 -4 Solve 𝑥 2 +6𝑥+6=0 By graphing.

Example 4: Approximate Roots with a table 7 6 5 4 3 2 1 -3 -2 -1 8 -4 Solve 𝑥 2 −4𝑥+2=0 By graphing. 𝑥= − −4 2 1 𝑥=2 𝑥=0.6, 3.4 𝒙 𝒚 2 1 −1 −2 3 4

Example 5: Approximate Roots with a Calculator MODEL ROCKETS Consuela built a model rocket for her science project. The equation ℎ=−16 𝑡 2 +250𝑡 models the flight of the rocket launched from ground level at a velocity of 250 feet per second, where ℎ is the height of the rocket in feet after 𝑡 seconds. Approximately how long was Consuela’s rocket in the air?

Example 5: Approximate Roots with a Calculator MODEL ROCKETS Consuela built a model rocket for her science project. The equation ℎ=−16 𝑡 2 +250𝑡 models the flight of the rocket launched from ground level at a velocity of 250 feet per second, where ℎ is the height of the rocket in feet after 𝑡 seconds. Approximately how long was Consuela’s rocket in the air? 15.6 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

Transformations of Quadratic Functions 9-3 Notes for Algebra 1 Transformations of Quadratic Functions

9-3 pg. 569 8-28, 42-63(x3)

Transformation Changes the position of size of a figure. 𝑓 𝑥 = 𝑥 2 →𝑔 𝑥 =−𝑎 𝑥+ℎ 2 +𝑘

Translation (A slide) moves a figure up, down, left or right. 𝑓 𝑥 = 𝑥 2 →𝑔 𝑥 = 𝑥+ℎ 2 +𝑘 Vertical Translation: 𝑘>0 𝑘 𝑢𝑛𝑖𝑡𝑠 𝑢𝑝 𝑘<0 ( 𝑘 𝑢𝑛𝑖𝑡𝑠 𝑑𝑜𝑤𝑛) Horizontal Translation: ℎ>0 ℎ 𝑢𝑛𝑖𝑡𝑠 𝑟𝑖𝑔ℎ𝑡 ℎ<0 ℎ 𝑢𝑛𝑖𝑡𝑠 𝑙𝑒𝑓𝑡

Example 1: Describe and Graph Translations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 =10+ 𝑥 2 2.) g 𝑥 = 𝑥 2 −8

Example 1: Describe and Graph Translations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 =10+ 𝑥 2 2.) g 𝑥 = 𝑥 2 −8 translated 10 units up translated 8 units down

Example 2: Horizontal Translations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 = 𝑥+1 2 2.) g 𝑥 = 𝑥−4 2

Example 2: Horizontal Translations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 = 𝑥+1 2 2.) g 𝑥 = 𝑥−4 2 translated 1 unit left translated 4 units right

Example 3: Horizontal and Vertical Translations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 = 𝑥+1 2 +1 2.) g 𝑥 = 𝑥−2 2 +6

Example 3: Horizontal and Vertical Translations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 = 𝑥+1 2 +1 2.) g 𝑥 = 𝑥−2 2 +6 shifted left 1 unit and up 1 shifted right 2 units and up 6

Dilation Makes a graph narrower or wider. 𝑓 𝑥 = 𝑥 2 →𝑔 𝑥 =𝑎 𝑥 2 Stretched vertically: 𝑎 >1 Compressed Vertically: 0< 𝑎 <1

Example 4: Describe and Graph Dilations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) d 𝑥 = 1 3 𝑥 2 2.) m 𝑥 =2 𝑥 2 +1

Example 4: Describe and Graph Dilations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) d 𝑥 = 1 3 𝑥 2 2.) m 𝑥 =2 𝑥 2 +1 Vertically compressed Vertically Stretched and shifted up 1 unit

Reflection A flip across a line. 𝑓 𝑥 = 𝑥 2 →𝑔 𝑥 =− 𝑥 2 Flip over a line.

Example 5: Describe and Graph Transformations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 =−3 𝑥 2 +1 2.) g 𝑥 = 1 5 𝑥 2 −7

Example 5: Describe and Graph Transformations Describe how the graph of each function is related to the graph of 𝑓 𝑥 = 𝑥 2 1.) g 𝑥 =−3 𝑥 2 +1 2.) g 𝑥 = 1 5 𝑥 2 −7 reflected in the x-axis Compressed vertically Stretched vertically shifted down 7 units shifted up 1 unit

Example 6: Identify an Equation for a Graph Which is an equation for the function shown in the graph? a.) 𝑦= 1 3 𝑥 2 −2 b.) 𝑦=3 𝑥 2 +2 c.) 𝑦=− 1 3 𝑥 2 +2 d.) 𝑦=−3 𝑥 2 −2

Example 6: Identify an Equation for a Graph Which is an equation for the function shown in the graph? a.) 𝑦= 1 3 𝑥 2 −2 b.) 𝑦=3 𝑥 2 +2 c.) 𝑦=− 1 3 𝑥 2 +2 d.) 𝑦=−3 𝑥 2 −2

Solving Quadratic equations by Completing the Square 9-4 Notes for Algebra 1 Solving Quadratic equations by Completing the Square

9-4 pg. 577 10-30, 37-42, 77-81o

Completing the Square Steps for completing the square of any quadratic expression. 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 1.) Make sure the leading coefficient (𝑎) is one. 2.) Get the 𝑎𝑥 2 and 𝑏𝑥 on one side and the 𝑐 term on a side by itself. 3.) Add 𝑏 2 2 to both sides of the equation (Then factor and simplify) 4.) Solve for 𝑥.

Example 1: Complete the Square 1.) Find the value of 𝑐 that makes 𝑥 2 −12𝑥+𝑐 a perfect square trinomial.

Example 1: Complete the Square 1.) Find the value of 𝑐 that makes 𝑥 2 −12𝑥+𝑐 a perfect square trinomial. 36

Example 2: Solve an Equation by completing the square 𝑥 2 +6𝑥+5=12

Example 2: Solve an Equation by completing the square 𝑥 2 +6𝑥+5=12 −7, 1

Example 3: Equation with 𝑎≠1 −2𝑥 2 +36𝑥−10=24

Example 3: Equation with 𝑎≠1 −2𝑥 2 +36𝑥−10=24 17, 1

Example 4: Use a Graph of a Quadratic Function CANOEING Suppose the rate of flow of an 80-foot-wide river is given by the equation 𝑟=−0.01 𝑥 2 +0.8𝑥, where 𝑟 is the rate in miles per hour and 𝑥 is the distance from the shore in feet. Joel does not want to paddle his canoe against a current that is faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour?

Example 4: Use a Graph of a Quadratic Function CANOEING Suppose the rate of flow of an 80-foot-wide river is given by the equation 𝑟=−0.01 𝑥 2 +0.8𝑥, where 𝑟 is the rate in miles per hour and 𝑥 is the distance from the shore in feet. Joel does not want to paddle his canoe against a current that is faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour? Up to 7 ft. from either bank NOTE: Then solutions of the equation are about 7 ft. and about 73 ft. Since the river is 80 ft. wide, 80−73=7. Both ranges are within 7 ft. of one bank or the other.

Solving Quadratic Equations by Using the Quadratic Formula 9-5 Notes for Algebra 1 Solving Quadratic Equations by Using the Quadratic Formula

9-5 pg. 587 16-40

Quadratic Formula 𝑎𝑥 2 +𝑏𝑥+𝑐=0 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎

Example 1: Use the Quadratic Formula 𝑥 2 −2𝑥=35

Example 1: Use the Quadratic Formula 𝑥 2 −2𝑥=35 −5, 7

Example 2: Use the Quadratic Formula 1.) 2𝑥 2 −2𝑥−5=0 2.) 5𝑥 2 −8𝑥=4

Example 2: Use the Quadratic Formula 1.) 2𝑥 2 −2𝑥−5=0 −1.2, 2.2 2.) 5𝑥 2 −8𝑥=4 −0.4, 2

Example 3: Solve Quadratic Equations Using Different Methods 3𝑥 2 −5𝑥=12

Example 3: Solve Quadratic Equations Using Different Methods 3𝑥 2 −5𝑥=12 − 4 3 , 3

Discriminant 𝑏 2 −4𝑎𝑐 Used to determine the number of real solutions of a quadratic equation. 𝑏 2 −4𝑎𝑐= Negative number (NO real number solutions) 𝑏 2 −4𝑎𝑐= Positive number (two solutions) 𝑏 2 −4𝑎𝑐= Zero (one solution)

Example 4: Use the Discriminant State the value of the discriminant of 3𝑥 2 +10𝑥=12. Then determine the number of real solutions of the equation. 244;2 real solutions

Analyzing Functions with Successive Differences 9-6 Notes for Algebra 1 Analyzing Functions with Successive Differences

9-6 pg. 593 14-25

Identify Functions Linear Function: 𝑦=𝑚𝑥+𝑏 Quadratic Function: 𝑦=𝑎𝑥 2 +𝑏𝑥+𝑐 Exponential Function: 𝑦=𝑎 𝑏 𝑥

Example 1: Choose a Model Using Graphs 10 9 8 7 6 5 4 3 2 1 -2 -1 1.) 1, 2 2, 5 3, 6 4, 5 5, 2

Example 1: Choose a Model Using Graphs 10 9 8 7 6 5 4 3 2 1 -2 -1 1.) 1, 2 2, 5 3, 6 4, 5 5, 2 Quadratic

Example 1: Choose a Model Using Graphs 2.) −1, 6 0, 2 1, 2 3 2, 2 9 .

Example 1: Choose a Model Using Graphs 10 9 8 7 6 5 4 3 2 1 -5 -4 -3 -2 -1 2.) −1, 6 0, 2 1, 2 3 2, 2 9 Exponential

Example 2: Choose a Model Using Differences or Ratios Look for a pattern in each table of values to determine which kind of model best describes the data. 1.) 2.) 𝒙 −2 −1 1 2 𝒚 3 5 7 𝒙 −2 −1 1 2 𝒚 36 12 4 4 3 4 9

Example 2: Choose a Model Using Differences or Ratios Look for a pattern in each table of values to determine which kind of model best describes the data. 1.) 2.) Linear Exponential 𝒙 −2 −1 1 2 𝒚 3 5 7 𝒙 −2 −1 1 2 𝒚 36 12 4 4 3 4 9

Example 3: Write an Equation Determine which model best describes the data. Then write an equation for the function that models the data 1.) 𝒙 1 2 3 4 𝒚 −1 −8 −64 −512 −4096

Example 3: Write an Equation Determine which model best describes the data. Then write an equation for the function that models the data 1.) Exponential, 𝑦=− 8 𝑥 𝒙 1 2 3 4 𝒚 −1 −8 −64 −512 −4096

Example 4: Write an Equation for Real-World Situation KARATE The table shows the number of children enrolled in beginner’s karate class for four consecutive years. Determine which model best represents the data. Then write a function that models that data. 1.) 𝒙 1 2 3 4 𝒚 −1 −8 −64 −512 −4096

Example 4: Write an Equation for Real-World Situation KARATE The table shows the number of children enrolled in beginner’s karate class for four consecutive years. Determine which model best represents the data. Then write a function that models that data. 1.) Linear, 𝑦=3𝑥+8 𝒙 1 2 3 4 𝒚 −1 −8 −64 −512 −4096

9-7 Notes for Algebra 1 Special Function

9-7 pg. 601 9-14, 17-30

Step Function A series of line segments called a piecewise-linear function.

Greatest integer function 𝑓 𝑥 = 𝑥 Disjointed line segments 𝐷−𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅−𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠

Example 1: Greatest Integer Function 6 5 4 3 2 1 -5 -4 -3 -2 -1 Graph 𝑓 𝑥 = 𝑥−2 State the domain and Range.

Example 1: Greatest Integer Function 6 5 4 3 2 1 -5 -4 -3 -2 -1 Graph 𝑓 𝑥 = 𝑥−2 State the domain and Range. 𝐷−𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅−𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠

Example 2: Greatest Integer Function TAXI A taxi company charges a fee For waiting at a rate of $0.75 per Minute or any fraction thereof. Draw a graph that represents this situation Cost ($) 5.25 4.50 3.75 3.00 2.25 1.50 0.75 1 2 3 4 5 6 7 Minutes

Example 2: Greatest Integer Function TAXI A taxi company charges a fee For waiting at a rate of $0.75 per Minute or any fraction thereof. Draw a graph that represents this situation Cost ($) 5.25 4.50 3.75 3.00 2.25 1.50 0.75 1 2 3 4 5 6 7 Minutes

Example 3: Absolute Value Function 10 9 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 Graph 𝑓 𝑥 = 2𝑥+2 State the domain and Range.

Example 3: Absolute Value Function 10 9 8 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 Graph 𝑓 𝑥 = 2𝑥+2 State the domain and Range. 𝐷−𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅−𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠

Example 4: Piecewise-Defined Function 6 5 4 3 2 1 -5 -4 -3 -2 -1 Graph 𝑓 𝑥 = −𝑥 𝑖𝑓 𝑥<0 −𝑥+2 𝑖𝑓 𝑥≥0 State the domain and Range.

Example 4: Piecewise-Defined Function 6 5 4 3 2 1 -5 -4 -3 -2 -1 Graph 𝑓 𝑥 = −𝑥 𝑖𝑓 𝑥<0 −𝑥+2 𝑖𝑓 𝑥≥0 State the domain and Range. 𝐷−𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅−𝑎𝑙𝑙 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠